Math 221 – F07
Activities, page 0
Activities for Use in Math 221
Compiled by R. Pruim
Fall, 2007
Math 221 – F07
Activities, page 1
Fill-In the Digits
1. Replace each letter with a digit so that the following sum is correct. Repeated letters must be
replaced by the same digit each time. Different letters must be replaced by different digits.
+
S
F
U
U
N
N
S
W
I
M
#$
2. 1–9 Triangle Puzzle
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a) Using each of the digits 1 through 9 exactly once, fill in the circles of a triangular arrangement similar to the one above so that the sums of the four digits along each side is the
same.
b) Find as many different solutions as you can in which the side sum is 20. (How should you
define “different”? Are some solutions more different than others?)
c) Can you find a solution in which the sum on each side is 15? 18? 21? 22?
3. 3-Digit Sums
Using each of the digits 1 through 9 exactly once, fill in the boxes below so that the sum is
correct:
+
Math 221 – F07
Activities, page 2
Some Problems
1. Ron’s Recycle Shop was started when Ron bought a used paper-shredding machine. Business
was good, so Ron bought a new shredding machine. The old machine could shred a truckload
of paper in 4 hours. The new machine could shred the same truckload of paper in only 2 hours.
How long will it take to shred a truckload of paper if Ron runs both shredders at the same
time?
2. One day my son Jason (at the time a first- or second-grader) made an observation and had a
question.
• The Observation: “If you have two numbers to add together and you take one away from
one number and give it to the other number they still add to the same thing.” (Example:
11 + 8 = 10 + 9.)
• The Question: “Does the same thing happen when you multiply two numbers?”
In my usual manner I answered his question by saying “What do you think?” And he proceeded
to think about the problem (and explain to me what he was thinking about the problem) until
he was satisfied that he understood the situation.
What can you learn about this observation and question?
3. At a local neighborhood festival, there was a tug-of-war competition.
• In round 1, one side had four acrobats, each equally strong, and the other side had five
neighborhood grandmas, also each equally strong. The result was dead even.
• In round 2, one side had Ivan, a dog, and the other side had two grandmas and one
acrobat. Again it was a draw.
• In round 3, Ivan and three grandmas went up agains four acrobats.
Which side won round 3?
4. A trapezoid is a four-sided figure with one pair of opposite sides parallel. Draw a trapezoid
that has an area of 36 square units.
5. Here is a pattern exploration exercise called Start and Jump Numbers. It goes like this. Pick a
start number and a jump number. Then make a list of numbers starting with the start number
and increasing by the jump number each time. For example, if the start number is 3 and the
jump number is 5, then the list starts 3, 8, 13, . . ..
Your task is to examine this list of numbers (using 3 as the start number and 5 as the jump
number) and to find as many patterns as you possibly can. You will probably want to start by
writing down the list up to numbers of about 130 or so.
Math 221 – F07
Activities, page 3
Procedures and Concepts
3 1
÷ ? Show how to compute it.
4 2
1. What is
2. Write a story problem for which
solution.
3
1
÷ would be the appropriate computation to find the
4
2
3. Find the area of the triangle below:
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5
"
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"
"
8
4. Why does your method work in problem 3?
5. Find 435 ÷ 18 without using a calculator.
6. In the following partial computation, why do we “bring down the 5”?
2
1 8 ) 4 3 5
3 6
7 5
7. In school you were probably taught that “division by zero is not allowed”. Why not?
Math 221 – F07
Activities, page 4
Some Problems to Solve
Work on the problems on this sheet in two phases:
1. First read through all the problems and make a list of ways you might attempt to solve the
problem. (Keep in mind the heuristics we have seen.) Jot down some notes for each problem,
then move to the next one.
2. Then go back and (attempt to) solve the problems. For each of the problems, explain the
reasoning you use to solve the problem. Whether you solve the problem completely or not, keep
a list of things you tried, patterns you observed, conjectures you made, facts you discovered,
etc. Be sure to note any problem solving strategies you used that did not occur to you when
you first read the problem (before working on it).
You may work the problems in any order.
1. What is the last digit of 399 ?
2. How many rectangles (of any size) are there in the picture below?
3. Jan is having a party, to which she invited 24 people. She plans to serve the meal on card
tables, arranged in a long rectangle, with each table pushed up against the next. If the card
tables are only large enough to seat one person on a side, how many tables will Jan need?
4. How many ways are there to cover the large rectangle below using “dominoes”. (A dominoe is
a rectangle that is 1 square wide and 2 squares tall.)
Dominoes:
[Hints: That large rectangle is pretty large, perhaps you should try a some smaller examples
first. How can you use smaller examples to help you with the original problem?]
5. A pail with 40 marbles in it weighs 175 grams. The same pail with 20 marbles in it weighs 95
grams. How much does the pale weigh alone? How much does one marble weigh alone?
6. How many ways are there to make change for 27 cents using (any number of) pennies, nickels,
dimes and quarters.
Math 221 – F07
Activities, page 5
Some More Problems
1. What is the last digit of 1399 ?
2. A regular pentagon is a figure with five equal (straight) sides and five equal angles. How many
diagonals are there in a regular pentagon? (The diagonals are the line segments that connect
one corner with another but are not sides. All together they look like a star.)
Can you generalize this for other polygons?
3. How many ways are there to make a path connecting adjacent letters in the diagram below so
that the path is labeled with the alphabet (each letter exactly once and in order)?
A
ABA
ABCBA
ABCDCBA
ABCDEDCBA
ABCDEFEDCBA
ABCDEFGFEDCBA
ABCDEFGHGFEDCBA
ABCDEFGHIHGFEDCBA
ABCDEFGHIJIHGFEDCBA
ABCDEFGHIJKJIHGFEDCBA
ABCDEFGHIJKLKJIHGFEDCBA
ABCDEFGHIJKLMLKJIHGFEDCBA
ABCDEFGHIJKLMNMLKJIHGFEDCBA
ABCDEFGHIJKLMNONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRSRQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRSTSRQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRSTUTSRQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRSTUVUTSRQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRSTUVWVUTSRQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRSTUVWXWVUTSRQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRSTUVWXYXWVUTSRQPONMLKJIHGFEDCBA
ABCDEFGHIJKLMNOPQRSTUVWXYZYXWVUTSRQPONMLKJIHGFEDCBA
4. There are 20 people in a room. If each person shakes hands with each other person exactly
once, how many handshakes occur?
5. How many squares (of any size) are there on an 8 × 8 checkerboard?